COMPLEX routines for (real) symmetric tridiagonal matrix
cstedc
USAGE:
work, rwork, iwork, info, d, e, z = NumRu::Lapack.cstedc( compz, d, e, z, [:lwork => lwork, :lrwork => lrwork, :liwork => liwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
* Purpose
* =======
*
* CSTEDC computes all eigenvalues and, optionally, eigenvectors of a
* symmetric tridiagonal matrix using the divide and conquer method.
* The eigenvectors of a full or band complex Hermitian matrix can also
* be found if CHETRD or CHPTRD or CHBTRD has been used to reduce this
* matrix to tridiagonal form.
*
* This code makes very mild assumptions about floating point
* arithmetic. It will work on machines with a guard digit in
* add/subtract, or on those binary machines without guard digits
* which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
* It could conceivably fail on hexadecimal or decimal machines
* without guard digits, but we know of none. See SLAED3 for details.
*
* Arguments
* =========
*
* COMPZ (input) CHARACTER*1
* = 'N': Compute eigenvalues only.
* = 'I': Compute eigenvectors of tridiagonal matrix also.
* = 'V': Compute eigenvectors of original Hermitian matrix
* also. On entry, Z contains the unitary matrix used
* to reduce the original matrix to tridiagonal form.
*
* N (input) INTEGER
* The dimension of the symmetric tridiagonal matrix. N >= 0.
*
* D (input/output) REAL array, dimension (N)
* On entry, the diagonal elements of the tridiagonal matrix.
* On exit, if INFO = 0, the eigenvalues in ascending order.
*
* E (input/output) REAL array, dimension (N-1)
* On entry, the subdiagonal elements of the tridiagonal matrix.
* On exit, E has been destroyed.
*
* Z (input/output) COMPLEX array, dimension (LDZ,N)
* On entry, if COMPZ = 'V', then Z contains the unitary
* matrix used in the reduction to tridiagonal form.
* On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
* orthonormal eigenvectors of the original Hermitian matrix,
* and if COMPZ = 'I', Z contains the orthonormal eigenvectors
* of the symmetric tridiagonal matrix.
* If COMPZ = 'N', then Z is not referenced.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1.
* If eigenvectors are desired, then LDZ >= max(1,N).
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
* If COMPZ = 'N' or 'I', or N <= 1, LWORK must be at least 1.
* If COMPZ = 'V' and N > 1, LWORK must be at least N*N.
* Note that for COMPZ = 'V', then if N is less than or
* equal to the minimum divide size, usually 25, then LWORK need
* only be 1.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal sizes of the WORK, RWORK and
* IWORK arrays, returns these values as the first entries of
* the WORK, RWORK and IWORK arrays, and no error message
* related to LWORK or LRWORK or LIWORK is issued by XERBLA.
*
* RWORK (workspace/output) REAL array, dimension (MAX(1,LRWORK))
* On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
*
* LRWORK (input) INTEGER
* The dimension of the array RWORK.
* If COMPZ = 'N' or N <= 1, LRWORK must be at least 1.
* If COMPZ = 'V' and N > 1, LRWORK must be at least
* 1 + 3*N + 2*N*lg N + 3*N**2 ,
* where lg( N ) = smallest integer k such
* that 2**k >= N.
* If COMPZ = 'I' and N > 1, LRWORK must be at least
* 1 + 4*N + 2*N**2 .
* Note that for COMPZ = 'I' or 'V', then if N is less than or
* equal to the minimum divide size, usually 25, then LRWORK
* need only be max(1,2*(N-1)).
*
* If LRWORK = -1, then a workspace query is assumed; the
* routine only calculates the optimal sizes of the WORK, RWORK
* and IWORK arrays, returns these values as the first entries
* of the WORK, RWORK and IWORK arrays, and no error message
* related to LWORK or LRWORK or LIWORK is issued by XERBLA.
*
* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK.
* If COMPZ = 'N' or N <= 1, LIWORK must be at least 1.
* If COMPZ = 'V' or N > 1, LIWORK must be at least
* 6 + 6*N + 5*N*lg N.
* If COMPZ = 'I' or N > 1, LIWORK must be at least
* 3 + 5*N .
* Note that for COMPZ = 'I' or 'V', then if N is less than or
* equal to the minimum divide size, usually 25, then LIWORK
* need only be 1.
*
* If LIWORK = -1, then a workspace query is assumed; the
* routine only calculates the optimal sizes of the WORK, RWORK
* and IWORK arrays, returns these values as the first entries
* of the WORK, RWORK and IWORK arrays, and no error message
* related to LWORK or LRWORK or LIWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: The algorithm failed to compute an eigenvalue while
* working on the submatrix lying in rows and columns
* INFO/(N+1) through mod(INFO,N+1).
*
* Further Details
* ===============
*
* Based on contributions by
* Jeff Rutter, Computer Science Division, University of California
* at Berkeley, USA
*
* =====================================================================
*
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cstegr
USAGE:
m, w, z, isuppz, work, iwork, info, d, e = NumRu::Lapack.cstegr( jobz, range, d, e, vl, vu, il, iu, abstol, [:lwork => lwork, :liwork => liwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CSTEGR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO )
* Purpose
* =======
*
* CSTEGR computes selected eigenvalues and, optionally, eigenvectors
* of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
* a well defined set of pairwise different real eigenvalues, the corresponding
* real eigenvectors are pairwise orthogonal.
*
* The spectrum may be computed either completely or partially by specifying
* either an interval (VL,VU] or a range of indices IL:IU for the desired
* eigenvalues.
*
* CSTEGR is a compatability wrapper around the improved CSTEMR routine.
* See SSTEMR for further details.
*
* One important change is that the ABSTOL parameter no longer provides any
* benefit and hence is no longer used.
*
* Note : CSTEGR and CSTEMR work only on machines which follow
* IEEE-754 floating-point standard in their handling of infinities and
* NaNs. Normal execution may create these exceptiona values and hence
* may abort due to a floating point exception in environments which
* do not conform to the IEEE-754 standard.
*
* Arguments
* =========
*
* JOBZ (input) CHARACTER*1
* = 'N': Compute eigenvalues only;
* = 'V': Compute eigenvalues and eigenvectors.
*
* RANGE (input) CHARACTER*1
* = 'A': all eigenvalues will be found.
* = 'V': all eigenvalues in the half-open interval (VL,VU]
* will be found.
* = 'I': the IL-th through IU-th eigenvalues will be found.
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input/output) REAL array, dimension (N)
* On entry, the N diagonal elements of the tridiagonal matrix
* T. On exit, D is overwritten.
*
* E (input/output) REAL array, dimension (N)
* On entry, the (N-1) subdiagonal elements of the tridiagonal
* matrix T in elements 1 to N-1 of E. E(N) need not be set on
* input, but is used internally as workspace.
* On exit, E is overwritten.
*
* VL (input) REAL
* VU (input) REAL
* If RANGE='V', the lower and upper bounds of the interval to
* be searched for eigenvalues. VL < VU.
* Not referenced if RANGE = 'A' or 'I'.
*
* IL (input) INTEGER
* IU (input) INTEGER
* If RANGE='I', the indices (in ascending order) of the
* smallest and largest eigenvalues to be returned.
* 1 <= IL <= IU <= N, if N > 0.
* Not referenced if RANGE = 'A' or 'V'.
*
* ABSTOL (input) REAL
* Unused. Was the absolute error tolerance for the
* eigenvalues/eigenvectors in previous versions.
*
* M (output) INTEGER
* The total number of eigenvalues found. 0 <= M <= N.
* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*
* W (output) REAL array, dimension (N)
* The first M elements contain the selected eigenvalues in
* ascending order.
*
* Z (output) COMPLEX array, dimension (LDZ, max(1,M) )
* If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
* contain the orthonormal eigenvectors of the matrix T
* corresponding to the selected eigenvalues, with the i-th
* column of Z holding the eigenvector associated with W(i).
* If JOBZ = 'N', then Z is not referenced.
* Note: the user must ensure that at least max(1,M) columns are
* supplied in the array Z; if RANGE = 'V', the exact value of M
* is not known in advance and an upper bound must be used.
* Supplying N columns is always safe.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1, and if
* JOBZ = 'V', then LDZ >= max(1,N).
*
* ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
* The support of the eigenvectors in Z, i.e., the indices
* indicating the nonzero elements in Z. The i-th computed eigenvector
* is nonzero only in elements ISUPPZ( 2*i-1 ) through
* ISUPPZ( 2*i ). This is relevant in the case when the matrix
* is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
*
* WORK (workspace/output) REAL array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal
* (and minimal) LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,18*N)
* if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* IWORK (workspace/output) INTEGER array, dimension (LIWORK)
* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK. LIWORK >= max(1,10*N)
* if the eigenvectors are desired, and LIWORK >= max(1,8*N)
* if only the eigenvalues are to be computed.
* If LIWORK = -1, then a workspace query is assumed; the
* routine only calculates the optimal size of the IWORK array,
* returns this value as the first entry of the IWORK array, and
* no error message related to LIWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* On exit, INFO
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = 1X, internal error in SLARRE,
* if INFO = 2X, internal error in CLARRV.
* Here, the digit X = ABS( IINFO ) < 10, where IINFO is
* the nonzero error code returned by SLARRE or
* CLARRV, respectively.
*
* Further Details
* ===============
*
* Based on contributions by
* Inderjit Dhillon, IBM Almaden, USA
* Osni Marques, LBNL/NERSC, USA
* Christof Voemel, LBNL/NERSC, USA
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL TRYRAC
* ..
* .. External Subroutines ..
EXTERNAL CSTEMR
* ..
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cstein
USAGE:
z, ifail, info = NumRu::Lapack.cstein( d, e, w, iblock, isplit, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO )
* Purpose
* =======
*
* CSTEIN computes the eigenvectors of a real symmetric tridiagonal
* matrix T corresponding to specified eigenvalues, using inverse
* iteration.
*
* The maximum number of iterations allowed for each eigenvector is
* specified by an internal parameter MAXITS (currently set to 5).
*
* Although the eigenvectors are real, they are stored in a complex
* array, which may be passed to CUNMTR or CUPMTR for back
* transformation to the eigenvectors of a complex Hermitian matrix
* which was reduced to tridiagonal form.
*
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the tridiagonal matrix T.
*
* E (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of the tridiagonal matrix
* T, stored in elements 1 to N-1.
*
* M (input) INTEGER
* The number of eigenvectors to be found. 0 <= M <= N.
*
* W (input) REAL array, dimension (N)
* The first M elements of W contain the eigenvalues for
* which eigenvectors are to be computed. The eigenvalues
* should be grouped by split-off block and ordered from
* smallest to largest within the block. ( The output array
* W from SSTEBZ with ORDER = 'B' is expected here. )
*
* IBLOCK (input) INTEGER array, dimension (N)
* The submatrix indices associated with the corresponding
* eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
* the first submatrix from the top, =2 if W(i) belongs to
* the second submatrix, etc. ( The output array IBLOCK
* from SSTEBZ is expected here. )
*
* ISPLIT (input) INTEGER array, dimension (N)
* The splitting points, at which T breaks up into submatrices.
* The first submatrix consists of rows/columns 1 to
* ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
* through ISPLIT( 2 ), etc.
* ( The output array ISPLIT from SSTEBZ is expected here. )
*
* Z (output) COMPLEX array, dimension (LDZ, M)
* The computed eigenvectors. The eigenvector associated
* with the eigenvalue W(i) is stored in the i-th column of
* Z. Any vector which fails to converge is set to its current
* iterate after MAXITS iterations.
* The imaginary parts of the eigenvectors are set to zero.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= max(1,N).
*
* WORK (workspace) REAL array, dimension (5*N)
*
* IWORK (workspace) INTEGER array, dimension (N)
*
* IFAIL (output) INTEGER array, dimension (M)
* On normal exit, all elements of IFAIL are zero.
* If one or more eigenvectors fail to converge after
* MAXITS iterations, then their indices are stored in
* array IFAIL.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, then i eigenvectors failed to converge
* in MAXITS iterations. Their indices are stored in
* array IFAIL.
*
* Internal Parameters
* ===================
*
* MAXITS INTEGER, default = 5
* The maximum number of iterations performed.
*
* EXTRA INTEGER, default = 2
* The number of iterations performed after norm growth
* criterion is satisfied, should be at least 1.
*
* =====================================================================
*
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cstemr
USAGE:
m, w, z, isuppz, work, iwork, info, d, e, tryrac = NumRu::Lapack.cstemr( jobz, range, d, e, vl, vu, il, iu, nzc, tryrac, [:lwork => lwork, :liwork => liwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO )
* Purpose
* =======
*
* CSTEMR computes selected eigenvalues and, optionally, eigenvectors
* of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
* a well defined set of pairwise different real eigenvalues, the corresponding
* real eigenvectors are pairwise orthogonal.
*
* The spectrum may be computed either completely or partially by specifying
* either an interval (VL,VU] or a range of indices IL:IU for the desired
* eigenvalues.
*
* Depending on the number of desired eigenvalues, these are computed either
* by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
* computed by the use of various suitable L D L^T factorizations near clusters
* of close eigenvalues (referred to as RRRs, Relatively Robust
* Representations). An informal sketch of the algorithm follows.
*
* For each unreduced block (submatrix) of T,
* (a) Compute T - sigma I = L D L^T, so that L and D
* define all the wanted eigenvalues to high relative accuracy.
* This means that small relative changes in the entries of D and L
* cause only small relative changes in the eigenvalues and
* eigenvectors. The standard (unfactored) representation of the
* tridiagonal matrix T does not have this property in general.
* (b) Compute the eigenvalues to suitable accuracy.
* If the eigenvectors are desired, the algorithm attains full
* accuracy of the computed eigenvalues only right before
* the corresponding vectors have to be computed, see steps c) and d).
* (c) For each cluster of close eigenvalues, select a new
* shift close to the cluster, find a new factorization, and refine
* the shifted eigenvalues to suitable accuracy.
* (d) For each eigenvalue with a large enough relative separation compute
* the corresponding eigenvector by forming a rank revealing twisted
* factorization. Go back to (c) for any clusters that remain.
*
* For more details, see:
* - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
* to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
* Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
* - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
* Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
* 2004. Also LAPACK Working Note 154.
* - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
* tridiagonal eigenvalue/eigenvector problem",
* Computer Science Division Technical Report No. UCB/CSD-97-971,
* UC Berkeley, May 1997.
*
* Further Details
* 1.CSTEMR works only on machines which follow IEEE-754
* floating-point standard in their handling of infinities and NaNs.
* This permits the use of efficient inner loops avoiding a check for
* zero divisors.
*
* 2. LAPACK routines can be used to reduce a complex Hermitean matrix to
* real symmetric tridiagonal form.
*
* (Any complex Hermitean tridiagonal matrix has real values on its diagonal
* and potentially complex numbers on its off-diagonals. By applying a
* similarity transform with an appropriate diagonal matrix
* diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
* matrix can be transformed into a real symmetric matrix and complex
* arithmetic can be entirely avoided.)
*
* While the eigenvectors of the real symmetric tridiagonal matrix are real,
* the eigenvectors of original complex Hermitean matrix have complex entries
* in general.
* Since LAPACK drivers overwrite the matrix data with the eigenvectors,
* CSTEMR accepts complex workspace to facilitate interoperability
* with CUNMTR or CUPMTR.
*
* Arguments
* =========
*
* JOBZ (input) CHARACTER*1
* = 'N': Compute eigenvalues only;
* = 'V': Compute eigenvalues and eigenvectors.
*
* RANGE (input) CHARACTER*1
* = 'A': all eigenvalues will be found.
* = 'V': all eigenvalues in the half-open interval (VL,VU]
* will be found.
* = 'I': the IL-th through IU-th eigenvalues will be found.
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input/output) REAL array, dimension (N)
* On entry, the N diagonal elements of the tridiagonal matrix
* T. On exit, D is overwritten.
*
* E (input/output) REAL array, dimension (N)
* On entry, the (N-1) subdiagonal elements of the tridiagonal
* matrix T in elements 1 to N-1 of E. E(N) need not be set on
* input, but is used internally as workspace.
* On exit, E is overwritten.
*
* VL (input) REAL
* VU (input) REAL
* If RANGE='V', the lower and upper bounds of the interval to
* be searched for eigenvalues. VL < VU.
* Not referenced if RANGE = 'A' or 'I'.
*
* IL (input) INTEGER
* IU (input) INTEGER
* If RANGE='I', the indices (in ascending order) of the
* smallest and largest eigenvalues to be returned.
* 1 <= IL <= IU <= N, if N > 0.
* Not referenced if RANGE = 'A' or 'V'.
*
* M (output) INTEGER
* The total number of eigenvalues found. 0 <= M <= N.
* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*
* W (output) REAL array, dimension (N)
* The first M elements contain the selected eigenvalues in
* ascending order.
*
* Z (output) COMPLEX array, dimension (LDZ, max(1,M) )
* If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
* contain the orthonormal eigenvectors of the matrix T
* corresponding to the selected eigenvalues, with the i-th
* column of Z holding the eigenvector associated with W(i).
* If JOBZ = 'N', then Z is not referenced.
* Note: the user must ensure that at least max(1,M) columns are
* supplied in the array Z; if RANGE = 'V', the exact value of M
* is not known in advance and can be computed with a workspace
* query by setting NZC = -1, see below.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1, and if
* JOBZ = 'V', then LDZ >= max(1,N).
*
* NZC (input) INTEGER
* The number of eigenvectors to be held in the array Z.
* If RANGE = 'A', then NZC >= max(1,N).
* If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
* If RANGE = 'I', then NZC >= IU-IL+1.
* If NZC = -1, then a workspace query is assumed; the
* routine calculates the number of columns of the array Z that
* are needed to hold the eigenvectors.
* This value is returned as the first entry of the Z array, and
* no error message related to NZC is issued by XERBLA.
*
* ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
* The support of the eigenvectors in Z, i.e., the indices
* indicating the nonzero elements in Z. The i-th computed eigenvector
* is nonzero only in elements ISUPPZ( 2*i-1 ) through
* ISUPPZ( 2*i ). This is relevant in the case when the matrix
* is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
*
* TRYRAC (input/output) LOGICAL
* If TRYRAC.EQ..TRUE., indicates that the code should check whether
* the tridiagonal matrix defines its eigenvalues to high relative
* accuracy. If so, the code uses relative-accuracy preserving
* algorithms that might be (a bit) slower depending on the matrix.
* If the matrix does not define its eigenvalues to high relative
* accuracy, the code can uses possibly faster algorithms.
* If TRYRAC.EQ..FALSE., the code is not required to guarantee
* relatively accurate eigenvalues and can use the fastest possible
* techniques.
* On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
* does not define its eigenvalues to high relative accuracy.
*
* WORK (workspace/output) REAL array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal
* (and minimal) LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,18*N)
* if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* IWORK (workspace/output) INTEGER array, dimension (LIWORK)
* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK. LIWORK >= max(1,10*N)
* if the eigenvectors are desired, and LIWORK >= max(1,8*N)
* if only the eigenvalues are to be computed.
* If LIWORK = -1, then a workspace query is assumed; the
* routine only calculates the optimal size of the IWORK array,
* returns this value as the first entry of the IWORK array, and
* no error message related to LIWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* On exit, INFO
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = 1X, internal error in SLARRE,
* if INFO = 2X, internal error in CLARRV.
* Here, the digit X = ABS( IINFO ) < 10, where IINFO is
* the nonzero error code returned by SLARRE or
* CLARRV, respectively.
*
*
* Further Details
* ===============
*
* Based on contributions by
* Beresford Parlett, University of California, Berkeley, USA
* Jim Demmel, University of California, Berkeley, USA
* Inderjit Dhillon, University of Texas, Austin, USA
* Osni Marques, LBNL/NERSC, USA
* Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
*
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csteqr
USAGE:
info, d, e, z = NumRu::Lapack.csteqr( compz, d, e, z, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE CSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
* Purpose
* =======
*
* CSTEQR computes all eigenvalues and, optionally, eigenvectors of a
* symmetric tridiagonal matrix using the implicit QL or QR method.
* The eigenvectors of a full or band complex Hermitian matrix can also
* be found if CHETRD or CHPTRD or CHBTRD has been used to reduce this
* matrix to tridiagonal form.
*
* Arguments
* =========
*
* COMPZ (input) CHARACTER*1
* = 'N': Compute eigenvalues only.
* = 'V': Compute eigenvalues and eigenvectors of the original
* Hermitian matrix. On entry, Z must contain the
* unitary matrix used to reduce the original matrix
* to tridiagonal form.
* = 'I': Compute eigenvalues and eigenvectors of the
* tridiagonal matrix. Z is initialized to the identity
* matrix.
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input/output) REAL array, dimension (N)
* On entry, the diagonal elements of the tridiagonal matrix.
* On exit, if INFO = 0, the eigenvalues in ascending order.
*
* E (input/output) REAL array, dimension (N-1)
* On entry, the (n-1) subdiagonal elements of the tridiagonal
* matrix.
* On exit, E has been destroyed.
*
* Z (input/output) COMPLEX array, dimension (LDZ, N)
* On entry, if COMPZ = 'V', then Z contains the unitary
* matrix used in the reduction to tridiagonal form.
* On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
* orthonormal eigenvectors of the original Hermitian matrix,
* and if COMPZ = 'I', Z contains the orthonormal eigenvectors
* of the symmetric tridiagonal matrix.
* If COMPZ = 'N', then Z is not referenced.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1, and if
* eigenvectors are desired, then LDZ >= max(1,N).
*
* WORK (workspace) REAL array, dimension (max(1,2*N-2))
* If COMPZ = 'N', then WORK is not referenced.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: the algorithm has failed to find all the eigenvalues in
* a total of 30*N iterations; if INFO = i, then i
* elements of E have not converged to zero; on exit, D
* and E contain the elements of a symmetric tridiagonal
* matrix which is unitarily similar to the original
* matrix.
*
* =====================================================================
*
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